Murphy's Law for Schemes

Abstract: This paper aims at presenting the necessary tools to prove that a scheme of finite type over Z exhibits the same singularities as those which occur on a Grassmann variety. First, basic theory regarding the combinatorial objects matroids is presented. Some important examples for the remainder of the paper are given, which also serve to aid the reader in intuition and understanding of matroids. Basic algebraic geometry is presented, and the building blocks affine varieties, projective varieties and general varieties are introduced. These object are generalised in the following subsection as affine schemes and schemes, which are the central object of study in modern algebraic geometry. Important results from the theory of algebraic groups are shown in order to better understand the formulation and proof of the Gelfand–MacPherson theorem, which in turn is utilised, together with Mnëv’s universality theorem, to prove the main result of the paper.